The Uncertainty Principle

where
is some constant determined by the
precise definitions of ``duration'' in the time domain and
``bandwidth'' in the frequency domain.

If duration and bandwidth are defined as the ``nonzero interval,''
then we obtain
, which is not useful. This conclusion
follows immediately from the definition of the Fourier transform
and its inverse (§2.2).

More interesting definitions of duration and bandwidth are obtained
using the normalized second moments of the squared magnitude:

(B.61)

where

By the DTFTpower theorem (§2.3.8), we have
. Note that writing ``
'' and
``
'' is an abuse of notation, but a convenient one.
These duration/bandwidth definitions are routinely used in physics,
e.g., in connection with the Heisenberg uncertainty principle [59].Under these definitions, we have the following theorem
[202, p. 273-274]:

We have considered two lower bounds for the time-bandwidth product
based on two different definitions of duration in time. In the
opposite direction, there is no upper bound on time-bandwidth
product. To see this, imagine filtering an arbitrary signal with an
allpass filter.B.3 The allpass filter cannot affect
bandwidth
, but the duration
can be arbitrarily extended by
successive applications of the allpass filter.